Mastery in mathematics

Thank you to Caroline Clissold for this informative maths article. This month, Caroline takes a look at mastery.

Part of my role as an independent consultant is to work with teachers in schools. Currently our focus is planning a mastery curriculum where all children can succeed. I’m finding that teachers really like the fact that they can link measures and statistics into the fundamental areas of mathematics: number sense, additive reasoning and multiplicative reasoning. They think making these connections are really valuable. They also really like the fact that they can spend longer exploring these key areas of mathematics. They are all using the correct vocabulary from the start and have been delighted that their children are using it with understanding. They also enjoy working the children in mixed attainment groupings and are very quickly seeing the benefits of this. So it really is worth trying!

Mastery is all about ensuring that all children understand what we are teaching them. So it means teaching concepts at a slower pace and dealing with each aspect of that concept step by step. In the past we have had teaching programmes that encourage us to spend a week on one thing and a few days on another. Evidence shows that this does not give time to enable children to understand a topic fully. So when we return to a topic at a later date children appear not to remember what we have taught them previously. It's not that they don't remember it's more that they haven't had the time to be able to understand. We want children to master what we teach them and this happens through step by step teaching and lots of practice within the concept itself and different contexts.

We discussed the fact that practice is very important. Perfect practice makes perfect! We therefore need to be sure that we give the children appropriate practice. Working through some worksheets or textbooks that are widely used in this country, don’t give perfect practice. Children need variation or, intelligent practice, which allows them to see the connections in mathematics and also allows them to deepen their understanding. So practice needs to begin with basic practice when the children are developing their understanding of something for the first time. This could involve the children using manipulatives and visual representations and may not include much, if any, recording.

They could then move on to practicing which involves seeing patterns, for example, if adding, they might explore what happens if they add 23 + 7, 43 + 7, 63 + 7. What is the same about these? What is different? They then might go on to explore the links with, for example, length, mass, capacity, volume, money and time. They might practice using information from statistical data and then create their own and use these to generate statements and questions. For example, one teacher I worked with was concerned that her children hadn’t mastered their multiplication facts for three. So we designed some lessons where they interrogated and then created pictograms. Each symbol represented three. They then interrogated and created then bar charts where the divisions increased in threes.

As well as spending longer on topics it is important to take one step at a time, so that we don’t confuse children by trying to teach them too many steps. We also need to be clear about the progression to the written methods and provide an efficient path to these. Sometimes we teach too many methods which some children find difficult to understand. I was watching a lesson recently where subtraction was being introduced using partitioning like this:

I couldn’t help thinking that this was a complex procedure, so much to do!

They had also been given manipulatives (Dienes) to help them. They weren’t using them because they didn’t know what to do with them. Probably the best approach would have been just to focus on using the manipulatives. They could practically take 25 away from 57 and just record 57 – 25 = 32. This would then give them opportunities to explore exchanging for calculations such as 63 – 49. Sometimes we over complicate!

For more information on mastery in mathematics, visit the NCTEM website or read their document on what mastery means for mathematics teaching.